Does the compositor affect image quality?
No, the compositor will not degrade rendered images.
The compositor works with 32bit float precision, in scene referred linear (meaning that there is way more information that what can be displayed in the monitor).
Of course, there could be intentional or unintentional degrading in the compositor caused by incorrectly connected nodes and/or poor judgement by the user.
A likely source of image degradation is the use of highly compressed images in lossy compression schemes (like jpeg or mpeg), in which the the original image will suffer corruption with each subsequent cycle of decompression and re-compression.
As for Color Banding, it would happen when the image is saved to 8 bit formats, as this method of encoding cannot resolve subtle color gradation with only 255 levels for each color. Areas of similar colors, and dark, almost monochromatic gradients, will suffer greatly.
To disguise banding in the rendered image, use dithering.
Note that even if the image is saved as 16 bit or 32 bit per channel, it is possible that it cannot be displayed at a better quality: Most monitors display only 8 bits per channel (and some even less than that). Banding then is inevitable (just pause any of your favorite movies during streaming and you will see what I'm talking about). Banding might disappear if the higher bit depth images are displayed properly in a 10 bit monitor or better (retina display or any professional grade monitor or video projector)